It begins with a boundary: A geometric view on probabilistically robust learning
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912050642419712 |
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| author | Bungert, Leon Trillos, Nicolás García Jacobs, Matt McKenzie, Daniel Nikolić, Đorđe Wang, Qingsong |
| author_facet | Bungert, Leon Trillos, Nicolás García Jacobs, Matt McKenzie, Daniel Nikolić, Đorđe Wang, Qingsong |
| contents | Although deep neural networks have achieved super-human performance on many classification tasks, they often exhibit a worrying lack of robustness towards adversarially generated examples. Thus, considerable effort has been invested into reformulating standard Risk Minimization (RM) into an adversarially robust framework. Recently, attention has shifted towards approaches which interpolate between the robustness offered by adversarial training and the higher clean accuracy and faster training times of RM. In this paper, we take a fresh and geometric view on one such method -- Probabilistically Robust Learning (PRL). We propose a mathematical framework for understanding PRL, which allows us to identify geometric pathologies in its original formulation and to introduce a family of probabilistic nonlocal perimeter functionals to rectify them. We prove existence of solutions to the original and modified problems using novel relaxation methods and also study properties, as well as local limits, of the introduced perimeters. We also clarify, through a suitable $Γ$-convergence analysis, the way in which the original and modified PRL models interpolate between risk minimization and adversarial training. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_18779 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | It begins with a boundary: A geometric view on probabilistically robust learning Bungert, Leon Trillos, Nicolás García Jacobs, Matt McKenzie, Daniel Nikolić, Đorđe Wang, Qingsong Machine Learning Analysis of PDEs Optimization and Control Although deep neural networks have achieved super-human performance on many classification tasks, they often exhibit a worrying lack of robustness towards adversarially generated examples. Thus, considerable effort has been invested into reformulating standard Risk Minimization (RM) into an adversarially robust framework. Recently, attention has shifted towards approaches which interpolate between the robustness offered by adversarial training and the higher clean accuracy and faster training times of RM. In this paper, we take a fresh and geometric view on one such method -- Probabilistically Robust Learning (PRL). We propose a mathematical framework for understanding PRL, which allows us to identify geometric pathologies in its original formulation and to introduce a family of probabilistic nonlocal perimeter functionals to rectify them. We prove existence of solutions to the original and modified problems using novel relaxation methods and also study properties, as well as local limits, of the introduced perimeters. We also clarify, through a suitable $Γ$-convergence analysis, the way in which the original and modified PRL models interpolate between risk minimization and adversarial training. |
| title | It begins with a boundary: A geometric view on probabilistically robust learning |
| topic | Machine Learning Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2305.18779 |